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A267967
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Integers n such that n^n is the sum of two nonzero squares while n is not.
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1
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30, 60, 70, 78, 102, 110, 120, 140, 150, 156, 174, 182, 190, 204, 210, 220, 222, 230, 238, 240, 246, 270, 280, 286, 300, 310, 312, 318, 330, 348, 350, 364, 366, 374, 380, 390, 406, 408, 420, 430, 438, 440, 444, 460, 470, 476, 480, 492, 494, 510, 518, 534, 540, 546, 550, 560
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OFFSET
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1,1
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COMMENTS
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Terms that are not divisible by 10 are 78, 102, 156, 174, 182, 204, 222, 238, 246, 286, 312, 318, 348, 364, 366, 374, 406, 408, 438, 444, 476, 492, 494, 518, 534, 546, 572, 574, 582, 598, 606, 624, 636, 638, 646, 654, 678, 696, 728, ...
If k^2 is the sum of 2 nonzero squares, (2*k)^(2*k) is. (2*k)^(2*k) = 2^(2*k) * k^(2*k) = (2^k)^2 * k^2 * k^(2*k-2) = (2^k)^2 * k^2 * (k^(k-1))^2. So if k^2 = a^2 + b^2, then (2*k)^(2*k) = (k^(k-1)*2^k*a)^2 + (k^(k-1)*2^k*b)^2. And if k^2 = a^2 + b^2 and k is not the sum of 2 nonzero squares, 2*k is not the sum of 2 nonzero squares. So 2 * A162592(n) appears in this sequence. Note that all terms appear as even numbers.
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LINKS
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EXAMPLE
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30 is a term because 30 is not the sum of 2 nonzero squares and 30^30 = 8609344200000000000000^2 + 11479125600000000000000^2.
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MATHEMATICA
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Select[Range@ 120, SquaresR[2, #] == 0 && Resolve[Exists[{a, b}, Reduce[#^# == (a^2 + b^2), {a, b}, Integers], a > b > 0]] &] (* Michael De Vlieger, Jan 24 2016 *)
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PROG
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(PARI) isA000404(n) = {for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2)); }
for(n=1, 1e3, if(isA000404(n^n) && !isA000404(n), print1(n, ", ")));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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